Number Systems, Logic Gates and Boolean Algebra
Domain 3 - Number systems, logic gates and Boolean algebra
Tests conversions among binary/denary/octal/hex, signed arithmetic, truth tables and simplification.
Core rules
- Positional value: each digit × base^position; group binary bits in 3 for octal, 4 for hex.
- Binary addition: 1+1=10 with carry; subtraction may use two's complement.
- Two's complement: invert bits and add 1; fixed width determines range, e.g. 8-bit −128 to +127.
- Gates: NOT, AND, OR, NAND, NOR, XOR; a truth table lists all input combinations.
- Boolean laws: identity, complement, distributive; De Morgan: (A·B)' = A'+B' and (A+B)' = A'·B'.
Worked example
Convert 101101₂ to denary and hex, showing reasoning. Denary: positions 5 to 0 give 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1 = 45. Hex: group from the right as 0010 1101 = 2D. Check: 2×16 + 13 = 45. So 101101₂ = 45₁₀ = 2D₁₆. For simplification, F = A·B + A·B' = A(B+B') = A·1 = A; this uses distribution and complement.
Traps
Do not group hex from the left; pad the leftmost group. In two's complement, the sign bit has negative weight. NAND is not AND; De Morgan changes both the operator and individual terms. Always show the law used.